Trigonometric substitution

In mathematics, trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions:

Note that the above step requires that "a" > 0 and cos("&theta;") > 0; we can choose the "a" to be the positive square root of "a"2; and we impose the restriction on "&theta;" to be −&pi;/2 < "&theta;" < &pi;/2 by using the arcsin function.

For a definite integral, one must figure out how the bounds of integration change. For example, as "x" goes from 0 to "a"/2, then sin(&theta;) goes from 0 to 1/2, so &theta; goes from 0 to &pi;/6. Then we have

:int_0^{a/2}frac{dx}{sqrt{a^2-x^2=int_0^{pi/6}d heta=frac{pi}{6}.

Some care is needed when picking the bounds. The integration above requires that −&pi;/2 < "&theta;" < &pi;/2, so "&theta;" going from 0 to &pi;/6 is the only choice. If we had missed this restriction, we might have picked "&theta;" to go from &pi; to 5&pi;/6, which would result in the negative of the result.